Limits in 3D Worksheet #1. Let f (x, y) = x2+y2 + 3 a) Discuss the continuity of f (x, y) by using the graph below to fill in the blanks (you should study this surface to see if you understand why it has this shape). GeoGebra 3D Calculator We can see that the function is continuous everywhere (there seem to be no breaks or holes). However, if we let x and y be. at the same time, we get an indeterminate expression (0/0). In other words, f (0,0) is So we can say that z = x2 + 1/2 Is actually at (0,0) after all. The surface seems to have only a "_ discontinuity) at (0,0). b) Now rewrite the function in #1 using the substitutions x = r cos 0 and y = r sin 0. See if you can simplify the limit until there is only the variable r (this is called a "polar substitution"). Don't forget to change the expression below the "lim" symbol also.#2. Let f (x, y)=y xetya. Verify the following limits (I gave you the answers, not just see if you can show work). Also write a brief sentence on your technique (how you evaluated it). a) lim xty 80 (x.y)-(25) x6+y3 189 Work: Now explain in a short paragraph how can you tell that f(x, y) is continuous at (2,5). b) xy (x.y) +(0,0) x*+y = DNE As always, we start with direct substitution, which gives us 0/0. However, just because you get 0/0 doesn't meant that the final answer is DIE. To convince ourselves that the answer really is DNE, we need to show that (at least) 2 paths ("hiking trails"), each passing through the origin, would arrive at two different outputs (z-values). To do this, let us pick any two curves that pass through (0,0). This can require some creativity, so I'll guide this first one. Let's try y = x and y = x2. These are different paths (as viewed from a bird's-eye view) yet they both pass through (0,0). You will try using the path y = x first. This means that you will replace all y variables with an x. Then try the limit. Show your work: Now try y = x2. Try the limit again. Show your work: Since we get more than one output (z), the limit is DNE. Next, explain this problem in your own words using a short paragraph. Make sure to compare and contrast this to how we showed limits were DIE in Chapter 2 back in September. Lastly, graph this one on the Geogebra 30 website and examine the graph carefully.The rest of these problems can be done with "polar substitution (see #1)." Your full set of options for substitution are listed below. Choose one or more of them to rewrite your problem with. r2 = x2 + yz 0 = Tan-'() x = r cos 0 y = r sin 0 Once you get the problem written in the variables r and/or 0, you may have to do a little bit more work. For example, if there is only r left, you could try L'Hopital's Rule. #3. Find the (xy)-(0.0) xty' xy2 #4. Find the (xy)-(0,0) x2+yz #5. Find the (0.0) lim (x2 + y?) In(x2 + y2). x7+()-1)2] #6. Find the lim cos 241 #7. Find the lim tan" x2+(y-1)=] x2+2 arctan (1" #8. Find the limit lim (x.y)-(2,0) In x2+ya#9. Let f (x, y) = sin(x2 +yz) x2+y/2 Then find the limits: a) lim (x,y)-(V2n/4,V2n/4) f(x, y) (this particular part 9a does NOT need a polar substitution. You can just use a direct substitution.) b) lim (x,y)-(0,0) f(x, y) (back to polar!) Now discuss the continuity of f (x, y)